Existence of Solutions Describing Accumulation in a Thin-Film Flow

Abstract

We consider a third order nonautonomous ODE that arises as a model of fluid accumulation in a two-dimensional thin-film flow driven by surface tension and gravity. With the appropriate matching conditions, the equation describes the inner structure of solutions around a stagnation point. In this paper we prove the existence of solutions that satisfy this problem. In order to prove the result we first transform the equation into a four-dimensional dynamical system. In this setting the problem consists of finding heteroclinic connections that are the intersection of a two-dimensional center-stable manifold and a three-dimensional center-unstable one. We then use a shooting argument that takes advantage of the information of the flow in the far-field; part of the analysis also requires the understanding of oscillatory solutions with large amplitude. The far-field is represented by invariant three-dimensional subspaces, and the flow on them needs to be understood; most of the necessary results in this regard ...